This paper shows how to use the shooting method, a classical numerical algorithm for solving boundary value problems, to compute the Riemannian distance on the Stiefel manifold $ \mathrm{St}(n,p) $, the set of $ n \times p $ matrices with orthonormal columns. The proposed method is a shooting method in the sense of the classical shooting methods for solving boundary value problems; see, e.g., Stoer and Bulirsch, 1991. The main feature is that we provide an approximate formula for the Fr\'{e}chet derivative of the geodesic involved in our shooting method. Numerical experiments demonstrate the algorithms' accuracy and performance. Comparisons with existing state-of-the-art algorithms for solving the same problem show that our method is competitive and even beats several algorithms in many cases.

翻译：本文展示了如何利用打靶法（一种经典的两点边值问题数值算法）计算Stiefel流形$\mathrm{St}(n,p)$上的黎曼距离。该流形由所有$n \times p$正交列矩阵构成。所提方法在本质上沿用了经典打靶法求解边值问题的思路（参见Stoer和Bulirsch, 1991）。其主要创新点在于给出了打靶过程中涉及测地线的Fréchet导数的近似计算公式。数值实验证明了算法的精度和性能。与求解相同问题的现有最先进算法相比，我们的方法具有竞争力，并且在许多情况下优于若干对比算法。